If some form of Computational Logic is the language of human thought, then the best place to look for it would seem to be inside our heads. But if we simply look at the structure and activity of our brains, it would be like looking at the hardware of a computer when we want to learn about its software. Or it would be like trying to do sociology by studying the movement of atomic particles instead of studying human interactions. Better, it might seem, just to use common sense and rely on introspection.

But introspection is notoriously unreliable. Wishful thinking can trick us into seeing what we want to see, instead of seeing what is actually there. The behavioural psychologists of the first half of the twentieth century were so suspicious of introspection that they banned it altogether.

Substitution for predicate letters is a syntactically intricate inference procedure that is derivable in standard systems of non-modal quantificational logic. It allows a new theorem to be deduced from a given theorem ϕ by replacing an atomic formula Pτ1 ··· τn within ϕ by another formula ψ, of arbitrary complexity, involving the terms τ1, …, τn. The intricacy comes in stating the restrictions that must be placed on free variables of ϕ and ψ for the substitution to be allowed.

Church [1956, p. 289] gives some historical notes on this rule, pointing out that it was inadequately stated in early works of Hilbert and Ackerman, Carnap, and Quine; and first correctly stated, but not in full generality, in Hilbert and Bernays's Grundlagen der Mathematik in 1934. Church himself calls the predicate letter P a functional variable, viewing it as a variable whose values are propositional functions of individuals. In non-modal logic, this means that P is interpreted as an n-ary function Un → {truth, falsehood} from individuals to truth-values.

Intuitively, a logic that is closed under substitution for predicate letters is one whose theorems represent “universal laws”, expressing properties that hold of all predicates, i.e. hold no matter what interpretation is given to the predicate letters, hence hold no matter what formulas are (correctly) substituted for them.

This book is about the possible-worlds semantic analysis of systems of logic that have quantifiers binding individual variables. Our approach is based on a notion of “admissible” model that places a restriction on which sets of worlds can serve as propositions. We show that admissible models provide semantic characterisations of a wide range of logical systems, including many for which the well-known model theory of Kripke [1963b] is incomplete. The key to this is an interpretation of quantification that takes into account the admissibility of propositions.

This is a subject that bristles with choices and challenges. Should terms be treated as rigid designators, or should their denotations vary from world to world? Should individual constants and variables be treated the same in this respect, or differently? Should each world have its own domain of existing individuals over which the quantifiable variables range, or should there be just a single domain of individuals? If there are varying domains, how should they relate to each other? Can any function from worlds to individuals be regarded as the “meaning” of some individual concept? Should an arbitrary mapping from individuals to propositions be admissible as a propositional function? Can we deductively axiomatise the class of valid formulas determined by each answer to these questions?

We now extend our language for quantified modal logic by adding a predicate symbol ≈ for an identity relation, allowing us to express assertions about the identity of individuals. A two-sorted language is developed, with one sort of term standing for individual concepts, represented as partial functions from worlds to individuals, and the other sort specialising to concepts that are rigid, i.e. have the same value in accessible worlds. The identity predicate allows the existence predicate ε to be defined, taking ετ to be the self-identity formula τ ≈ τ, whose corresponding proposition/truth set is the domain of the partial function interpreting τ. The use of admissibility is extended from propositions to individuals by requiring each model to have a designated set of admissible individual concepts, within which there is a designated set of admissible rigid ones.

We axiomatise the set of formulas that are valid in these models, using a new inference rule that allows deduction of assertions of non-existence (¬ετ). The logic characterised by Kripkean models is then treated separately. The final section of the chapter gives a semantic analysis of Russelian definite description terms ιx.ϕ in this context, and shows how to construct canonical models and axiomatisations for logics in languages that have these description terms as well as the identity predicate.

The main aim of this chapter is to set out a new kind of admissible model theory for the propositional relevant logic R and its quantified extension RQ. First we review the relational semantics for R of Routley and Meyer [1973], and its adaptation by Mares and Goldblatt [2006] to an admissible semantics for RQ. Then we introduce an alternative kind of structure, called a cover system, motivated by topological ideas about “local truth” from the Kripke-Joyal semantics for intuitionistic logic in topos theory. These are combined with a modelling of negation by a binary world-relation of orthogonality, or incompatibility, as in [Goldblatt 1974], and an operation of combination, or “fusion”, of worlds to interpret relevant implication. Characteristic model systems for R have an algebra Prop of admissible propositions, while those for RQ have a set PropFun of admissible propositional functions as well.

We then show that by conservatively adding an intuitionistic implication connective to R it is possible to characterise that logic by models in which all possible propositions are admissible. The prospects for a similar analysis of RQ are considered at the end.

Routley-Meyer Models for R

The subject of relevant logic (also known as relevance logic) is based on the view that an implication A → B can only be true if the meaning of A is relevant to the meaning of B.

According to A Course in Game Theory, “a game is a description of strategic interaction that includes the constraints on the actions that the players can take and the players' interests, but does not specify the actions that the players do take” [45, p. 2]. Classical game theory makes a distinction between strategic and extensive games. In a strategic game each player moves only once, and all the players move simultaneously. Strategic games model situations in which each player must decide his or her course of action once and for all, without being informed of the decisions of the other players. In an extensive game, the players take turns making their moves one after the other. Hence a player may consider what has already happened during the course of the game when deciding how to move.

We will use both strategic and extensive games in this book, but we consider extensive games first because how to determine whether a first-order sentence is true or false in a given structure can be nicely modeled by an extensive game. It is not necessary to finish the present chapter before proceeding. After reading the section on extensive games, you may skip ahead to Chapter 3. The material on strategic games will not be needed until Chapter 7.

Extensive games

In an extensive game, the players may or may not be fully aware of the moves made by themselves or their opponents leading up to the current position.